Force-dipole flow ($\mathcal{P}$)

The leading far field of many force-free microswimmers is an axisymmetric force dipole, or stresslet,

Two equal parallel axial point stresslets with positive strength $S$ and separation $\ell$ have relative velocity $S/(2\pi \mu {\ell }^2)$. Identical self-propulsion cancels in their relative motion. The solution is the displayed cubic separation law while the far-field approximation is valid.

The angular factor of an axial stresslet is linear in the symmetric traceless rank-two tensor $\mathbf{ee}-I/3$. Hence averaging a collection of directors amounts to averaging this tensor. For uniformly distributed in-plane directions, $⟨\mathbf{ee}⟩=(I-\mathbf{nn})/2$, so $⟨S(\mathbf{ee}-I/3)⟩=-(S/2)(\mathbf{nn}-I/3)$. The averaged director is normal to the plane and its signed strength is reversed and halved.

A stresslet whose centre follows a circle and whose director stays tangent has a leading far-field mean stresslet centred at the circle centre. Orientation averaging of an axisymmetric stresslet gives normal-axis strength $-S/2$. Thus a pusher microswimmer becomes a leading mean puller microswimmer, and conversely. The radius and traversal frequency enter higher-order spatial corrections or averaging time, but not the leading dipole strength.

Opposite points on the orbit have opposite displacement and the same stresslet director up to sign. Their paired fields are $[G(\mathbf{r}-R\mathbf{p},\mathbf{e})+G(\mathbf{r}+R\mathbf{p},\mathbf{e})]/2$, so odd displacement powers cancel. Since a stresslet decays as $r^{-2}$, the first correction to its far-field orbit average of a tangent stresslet is $O(S{R}^2/(\mu r^4))$. The finite-radius mean need not be an exact point stresslet.

Opposite forces $\pm F\mathbf{e}$ applied to the fluid at $\pm (a/2)\mathbf{e}$ have no net force or torque. Their leading Stokeslet difference is $-Fa(\mathbf{e}\cdot \nabla )[J(\mathbf{r})\mathbf{e}]$, equal to the axial stresslet with strength $S=Fa$. Positive $S$ drives outward flow on the axis and inward flow in the equatorial plane, corresponding to a pusher microswimmer; negative $S$ gives a puller microswimmer.

A pusher microswimmer generates an extensile force-dipole flow, expelling fluid along its swimming axis and drawing fluid inward from its sides. In the displayed convention it has $\mathcal{P}>0$.

A puller microswimmer generates a contractile force-dipole flow, drawing fluid inward along its swimming axis and expelling it sideways. In the displayed convention it has $\mathcal{P}<0$.

Reflecting a force dipole's position and normal orientation component across a flat clean free surface produces an image whose superposition enforces no penetration and zero tangential traction. Its flow at the swimmer causes both normal drift and reorientation.

For equal stresslets with height $h$ and axial separation $\ell \gg h$, the neighbour image doubles the leading axial repulsion. Its normal contribution is smaller than own-image attraction by order $(h/\ell {)}^3$, so leading height dynamics are unchanged. The approximation also neglects weak neighbour-induced orientation corrections.

A positive-strength parallel point stresslet below a flat shear-free impermeable plane is attracted by its identical reflected image. Its height satisfies $\dot{h}=-S/(32\pi \mu h^2)$ and formally reaches zero in finite time. Negative strength reverses the drift. Finite-body near-contact physics lies outside this singularity prediction.